Exploring dark sector parameters in light of neutron star temperatures
EExploring dark sector parameters in light of neutron startemperatures
Guey-Lin Lin ∗ and Yen-Hsun Lin
1, 2, † Institute of Physics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan Institute of Physics, Academia Sinica, Taipei 11529, Taiwan (Dated:)
Abstract
Neutron star (NS) as the dark matter (DM) probe has gained a broad attention recently, eitherfrom heating due to DM annihilation or its stability under the presence of DM. In this work, weinvestigate spin-1 / χ charged under the U (1) X in the dark sector. The massivegauge boson V of U (1) X gauge group can be produced in NS via DM annihilation. The producedgauge boson can decay into Standard Model (SM) particles before it exits NS, despite its tinycouplings to SM particles. Thus, we perform a systematic study on χ ¯ χ → V → χ ¯ χ → χV scattering is also considered. We assume the general frameworkthat both kinetic and mass mixing terms between V and SM gauge bosons are present. This allowsboth vector and axial-vector couplings between V and SM fermions even for m V (cid:28) m Z . Notably,the contribution from axial-vector coupling is not negligible when particles scatter relativistically.We point out that the above approaches to DM-induced NS heating are not yet adopted in recentanalyses. Detectabilities of the aforementioned effects to the NS surface temperature by the futuretelescopes are discussed as well. ∗ [email protected] † [email protected] a r X i v : . [ h e p - ph ] F e b . INTRODUCTION It has been widely accepted that one-fifth of the total energy of the Universe consists ofdark matter (DM). Though multidisciplinary strategies are employed to identify its essence,either from direct [1–10] or indirect detections [11–16], the nature of DM remains a puz-zle. The approach of using neutron star (NS) as the DM probe has been proposed fromthe heating effect due to DM [17–26], the NS instability caused by DM gravitational col-lapse [27–38] and gravitation wave emitted from the merger of binary NS admixed withDM [39–41]. Novel way of constraining long-lived particle through the NSs in the MilkyWay is also investigated recently [42]. In addition, DM self-interaction naturally arises invarious phenomenological models and was proposed to resolve many issues in the small-scalestructure, e.g. core-cusp, missing satellite, too-big-to-fail and diverse galactic rotation curve,see Ref. [43] for a review. Current astrophysical observations constrain DM self-interactioncross section σ χχ in the range [44–48]0 . g − ≤ σ χχ /m χ ≤
10 cm g − (1)where m χ is the DM mass.Without delving into details of model constructions, DM self-interaction can be under-stood phenomenologically as an exchange of a vector boson V or a scalar boson φ in the darksector (DS). Here φ is the dark Higgs responsible for a spontaneously symmetry breaking ofAbelian U (1) X in DS, and therefore the generation of V boson mass. Assuming DM χ is aspin-1 / U (1) X gauge coupling g d , DM self-interaction is inducedby L DM − DM = g d ¯ χγ µ χV µ and constrained by Eq. (1). Furthermore, V can mix with SMphoton and Z boson through kinetic [49–54] and mass mixing terms [55–57]. These mixingterms appear in the following Lagrangians: L gauge = − B µν B µν + 12 ε γ cos θ W B µν V µν − V µν V µν , (2) L mass = 12 m Z Z µ Z µ − ε Z m Z Z µ V µ + 12 m V V µ V µ , (3)where B µν ≡ ∂ µ B ν − ∂ ν B µ is the U (1) Y field strength in SM while ε γ and ε Z are the kineticand V − Z mass mixing parameters respectively. The electromagnetic (EM) and neutral-current (NC) interactions between V and SM fermions f resulting from mixing terms in2qs. (2) and (3) are given by L DS − SM = (cid:18) ε γ eJ EM µ + ˜ ε Z g cos θ W J NC µ (cid:19) V µ (4)where g is the SU (2) L coupling and J EM µ and J NC µ are SM electromagnetic and neutralcurrents, respectively. The coefficient ˜ ε Z is a linear combination of two mixing parametersand it reduces to ε Z for m V (cid:28) m Z . Its general expression is given in Appendix A.In this paper, we examine the effect of DM heating due to the above phenomenologicalsetup for a nearby 3 giga-year-old (Gyr-old) and isolated NS. The associated temperatureis around 100 K according to the standard cooling mechanism if there is no other heatingsource. Therefore, any temperature deviating from this benchmark value can be potentiallydue to DM annihilation in the star. Besides, the dark boson V can be produced via χ ¯ χ → V for m V < m χ . Thus, V decaying into a fermion pair inside the star is possible. This impliesthat the heating from χ ¯ χ → V cannot be ignored if the decay length of V is smaller thanthe star’s radius. Searching the nearby old and cold NS can improve our understanding onDM. The new dynamics emerging from the above phenomenological setup will be discussedin the following sections. For completeness, we also analyze the signal-to-noise ratio (SNR)in the James Webb Space Telescope (JWST) [58]. Future telescopes such as EuropeanExtremely Large Telescope (E-ELT) and Thirty-Meter Telescope (TMT) [59] will constrainDM properties with unprecedented sensitivities. In the following sections, we employ theNS mass M = 1 . M (cid:12) and and the radius R = 12 km. We also replace g d with α χ = g d / π and all equations are expressed in terms of natural units (cid:126) = c = k B = 1. II. DM CAPTURE AND NS TEMPERATURE
When a NS swipes through the space, the DM particles in the halo can scatter withthe baryons and leptons inside the star. Once DMs lost an appreciable fraction of kineticenergies, they will be gravitationally captured by the NS. This capture process has beeninvestigated extensively with contributions from neutrons, protons and leptons as well asrelativistic correction included in Refs. [60, 61]. In this paper, only neutron contribution tothe capture rate C c is considered. Contributions from other particle species are ignored dueto their small yields. The total DM number N χ in the star satisfies the differential equation dN χ dt = C c − C a N χ , (5)3here C a is the DM annihilation rate. Both coefficients C c and C a are well studied and theexpressions can be found in Refs. [22, 60, 61] and references therein. We do not reproducehere. Thus, the exact solution to Eq. (5) is obtained N χ = C c τ eq tanh( t/τ eq ) (6)where τ eq = 1 / √ C c C a is the equilibrium timescale. Once t > τ eq , dN χ /dt = 0 and N χ ( t >τ eq ) = (cid:112) C c /C a according to Eq. (5). The total annihilation rate at this stage only dependson the capture rate since Γ a = C a N χ = C c . Note that C c depends on σ χn and σ χn ≤ σ geom χn ≈ − cm where σ geom χn is the geometric cross section. In principle, the maximum capturerate is determined by C c ( σ geom χn ) = C geom c . Besides, when DM falls into the NS surface, itis accelerated up to 0 . c − . c . The non-relativistic (NR) limit for calculating σ χn is notapplicable. Furthermore one has to consider contributions from axial-vector coupling dueto V − Z mass mixing given by L mass . We have thoroughly included these effects. A briefdiscussion on how to compute σ χn in terms of relativistic kinematics is given in Appendix A.NS is known to suffer from eternal cooling due to neutrino and photon emissions. Withoutextra energy injection, the NS temperature drops until it releases all its heat. However, if SMparticles are produced due to DM annihilation in the star, these particles can become a heatsource and potentially prevent the star from inevitable cooling. Therefore, the evolution ofNS interior temperature T b is governed by the equation dT b dt = − (cid:15) ν − (cid:15) γ + (cid:15) χ c V , (7)where (cid:15) ν ≈ . × erg cm − s − ( T b / K) is the neutrino emissivity, (cid:15) γ ≈ . × erg cm − s − ( T b / K) . the photon emissivity, (cid:15) χ the DM emissivity that is respon-sible for the heating from DM annihilation and c V the NS heat capacity [17]. Addi-tionally, the surface temperature T s observed by a distant observer is related to T b by T s ≈ . × K ( g s / cm s − ) / ( T b / K) . where g s = GM/R ≈ . × cm s − accounts for the redshift correction from the star’s surface gravity. It is also pointed outthat when T b < T b and T s [22].During each annihilation, a pair of DMs release 2 m χ of energy in a form of SM particlesor dark bosons depending on which channel are kinematically allowed. The total energyreleasing rate by DM is E χ = 2 m χ Γ a (cid:80) i b i where b i is the branching ratio of a specific channel,eg. e ± , µ ± , τ ± or q ¯ q , and (cid:80) i b i ≤
1. Neutrino pair ν ¯ ν is also part of the annihilation channel4 a) (b) FIG. 1: DM heating from dark boson production χ ¯ χ → V . Left : V decays into SMparticles before it exits the star. Right : V is self-trapped due to multiple χV scatteringsand then decays.in the presence of V − Z mass mixing, but it cannot contribute to the heating. In addition tothe annihilation, DMs also lose kinetic energies E k to the star through the capturing process.This has been realized as the kinetic heating [20] with the rate K χ = C c E k = C c m χ ( γ − γ = 1 / √ − v is the Lorentz factor. Thus, DM emissivity (cid:15) χ is given by (cid:15) χ = E χ + K χ V , (8)where V is the NS volume. III. DECAYS OF DARK BOSON
Here we discuss the case of V produced by DM annihilation. V is usually producedin DM rich environment. If V can subsequently scatter off the surrounding DMs multipletimes, it could lose energy and be self-trapped. It then decays promptly as shown in Fig. 1b.However, such self-trapping effect is in general inefficient since χV scattering length (cid:96) χV ismuch larger than the thermal radius r th . Hence the scattering rate is much suppressed andirrelevant to the heating. We leave the detail discussions in Appendix C. Another trappingis due to the scattering between V and neutrons. On the other hand the relevant cross Even DM is not captured, energy deposition still occurs as long as χn scattering can happen. On theother hand, the kinetic heating effect from such uncaptured DM is relatively small and negligible in ourcalculation. ε Z and the scattering length is expected to bemuch larger than the NS radius. It is safe to omit this effect in our calculation as well.However, V can decay into other SM particles before it propagates to the surface as longas the decay length (cid:96) dec is shorter than R . See Fig. 1a. The decay length is given by (cid:96) dec = vγτ dec with v ≡ (cid:113) − m V /m χ the velocity of V and τ dec ≡ Γ − V the lifetime of V atrest where Γ V is the total decay width. Since V is produced on shell, we do not consider V decaying back to χ due to m V < m χ . The probability for V to convert into SM particlesafter a propagation distance r is F = 1 − e − r/(cid:96) dec . (9)We took r = R in the calculation. However, if neutrino is the decay product, it cannotbe considered as the heating source and must be subtracted. By examining the numericalresults for F , we found that V can decay before it exits the star in most of our interestedparameter space. This implies that χ ¯ χ → V also plays an important role in NS heating.See Appendix C for details. Generally speaking, NS contains muons and electrons that arein degenerate. To enable the decay V → e + e − , m V not only has to be heavier than 2 m e ,the final kinetic energy carried by e − must also exceed the electron chemical potential toprevent from Pauli blocking. This condition has been implemented in our study.Given the information in this section, we summarize that even when χ ¯ χ → V dominatesthe annihilation channel for m V < m χ , the heating effect is still efficient due to V decays.However, the self-trapping is generally unimportant due to (cid:96) χV (cid:29) r th in this paper. IV. IMPLICATION OF DM ON NS TEMPERATURE
In this section, we describe how NS surface temperature T s is affected by the DM annihi-lation. If (cid:15) χ is negligible, the standard cooling mechanism gives T s ≈
100 K for a 3-Gyr-oldNS. But when (cid:15) χ is large enough to counterbalance (cid:15) γ,ν , T s could remain in a relativelyhigher temperature. We present the numerical results of T s for both α χ = 1 and 0 .
01 inFigs. 2 and 3 respectively. The adjacent DM density around NS is assumed to be the sameas that of the solar system, ρ χ = 0 . / cm , since we aims for the nearby isolated NS.The DM mass scale is shown from 100 MeV to 10 MeV. Once m χ (cid:46)
100 MeV, all of theannihilation channels to fermions will be Pauli blocked except neutrinos. Nonetheless, thereis no upper limit for DM mass in NS. But heavier m χ results in lesser DM number den-6ity which makes the NS sensitivity worse. In addition, Refs. [62, 63] pointed out when m χ (cid:38) O (10 − α χ = 1, Fig. 2, unless specified otherwise. The conclusions can be applied to α χ = 0 . α χ is that the dark sector interactions are proportionalto α χ and DM-SM interactions are to α χ . The derivations of such features on the scatteringcross sections for all interactions are given in the appendices.The values for the parameter η ≡ ε γ /ε Z from top to bottom are 1 (combined, ε Z = ε γ (cid:54) = 0), 0 (pure V − Z mixing, ε γ = 0) and ∞ (pure kinetic mixing, ε Z = 0), respectively.From left to right, we have m V /m χ = 10 (heavy mediator), 1 (equal mass) and 0 . T s is indicated by the color bar placed on the right and the lowest temperatureis 100 K. Without annihilation, e.g. no anti-DM exists, solely kinetic heating can raise T s up to 1750 K. If DM annihilation is included, T s can maximally reach to 3100 K.Various constraints are also plotted, including XENON1T [8], XENON LDM (low massDM) based on the ionization [10] and of Migdal [9] effects, SIDM [44–48], SN1987A [67] andbeam dump experiments [64–66]. The parameter curve rendering DM annihilation crosssection at the thermal relic value 6 × − cm s − in the early Universe is plotted in greenon each figure. We have adopted the method given in Ref. [69] for computing the Sommerfeldenhancement factor. The DM relative velocity in the early Universe is taken to be c/
3. SeeAppendix B for details. Here we present the thermal relic cross section as a reference pointand refer the readers to Refs. [53, 70, 71] for detailed discussions. In addition, althoughthe captured DMs can have relatively large Sommerfeld enhancement due to low velocities, the enhanced σv only shortens the equilibrium timescale τ eq . When t (cid:29) τ eq , the totalannihilation rate only depends on the capture rate with Γ A = C c . NS is generally insensitiveto the Sommerfeld enhancement as long as DMs are in equilibrium. Assuming DMs are thermalized with the NS core where T χ = T b . Thus the mean velocity is about (cid:112) T χ /m χ . � � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � α χ = �� � � = �� � χ � η = � ������������ 〈σ � 〉 ���� ��������� � � � �� � � � �� � � � � �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � α χ = �� � � ~ � χ � η = � � � � � � � � � � �� � 〈 σ � 〉 ���� ��������� ���� ��� ��� � � � � � � � � �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � α χ = �� � � = ��� � χ � η = � ���� ��������� ���� ������ � � � � � � � � ������������������������ � � [ ⨯ �� � � ] �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � α χ = �� � � = �� � χ � η = � ������������ 〈σ � 〉 ���� ��������� � � � �� � � � �� � � � � �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � α χ = �� � � ~ � χ � η = � � � � � � � � � � �� � 〈 σ � 〉 ���� ��������� ���� ��� ��� � � � � �������� � � � � �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � α χ = �� � � = ��� � χ � η = � ���� ��������� ���� ������ � � � � ���� ���� � � � � ������������������������ � � [ ⨯ �� � � ] �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε γ α χ = �� � � = �� � χ � η = ∞ ������������ 〈σ � 〉 � � � �� � � � �� � � � � �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε γ α χ = �� � � ~ � χ � η = ∞ � � � � � � � � � �� � 〈 σ � 〉 � � � �� � � � �� � � � � ���������� � � � � �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε γ α χ = �� � � = ��� � χ � η = ∞ ���� ��� ��� � � � � ������ ���� � � � � ������������������������ � � [ ⨯ �� � � ] FIG. 2: NS surface temperature T s in the m χ − ε plane. We took the age of NS is 3 Gyrsand the lowest T s = 100 K without DM heating. All figures have α χ = 1 and η = ε γ /ε Z .From top to bottom, η = 1 ,
0, and ∞ . From left to right, m V /m χ = 10 , , .
1. Variousconstraints from XENON1T [8], XENON LDM [9, 10], SIDM [44–48], SN1987A [67, 68],beam dump experiments [64–66] and the parameter curve rendering thermal relic crosssection are shown as well.
A. Case for m V ≥ m χ When m V ≥ m χ , only χ ¯ χ → f ¯ f is allowed. A dip occurs on each plot in Fig. 2 withthis mass ordering. The resonant point is caused by the pole in ˜ ε Z given by Eq. (A5)8 � � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � α χ = ����� � � = �� � χ � η = � � � � �� � � � �� � � � � �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � α χ = ����� � � ~ � χ � η = � � � � � � � � � � �� � 〈 σ � 〉 ���� ��������� ���� ������ � � � � �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � α χ = ����� � � = ��� � χ � η = � � � � � � � � � � �� � 〈 σ � 〉 ���� ��������� ���� ������ � � � � � � � � ������������������������ � � [ ⨯ �� � � ] �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � α χ = ����� � � = �� � χ � η = � � � � �� � � � �� � � � � �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � α χ = ����� � � ~ � χ � η = � � � � � � � � � � �� � 〈 σ � 〉 ���� ��������� ���� ������ � � � � �������� �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � α χ = ����� � � = ��� � χ � η = � � � � � � � � � � �� � 〈 σ � 〉 ���� ��������� ���� ������ � � � � ���� ���� � � � � ������������������������ � � [ ⨯ �� � � ] �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε γ α χ = ����� � � = �� � χ � η = ∞ � � � � �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε γ α χ = ����� � � ~ � χ � η = ∞ ������������ 〈σ � 〉 � � � �� � � � �� � � � � �� �������� �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε γ α χ = ����� � � = ��� � χ � η = ∞ � � � � � � � � � �� � 〈 σ � 〉 ���� ��� ��� � � � � ���������� � � � � ������������������������ � � [ ⨯ �� � � ] FIG. 3: The same as Fig. 2 except α χ = 0 . m V = m Z with m Z the SM Z boson mass. In fact the value for ˜ ε Z at this point is − i ( ε Z + ε γ tan θ W ) m Z / Γ Z , which is enhanced by the factor m Z / Γ Z .Thus, the DM-neutron scattering cross section σ χn depends on ˜ ε Z and is proportional to σ χn ∝ α χ ˜ ε Z m V m χ m n ( m χ + m n ) min( ξ,
1) (10)in the NR limit. See Eq. (A12) for reference. The last term shows the suppression factor due In the numerical calculation, we used the general expression for σ χn , Eq. (A7), and the derivation is givenin the same appendix. Nonetheless, Eq. (10), or Eq. (A12), is simpler and suitable for our discussions inthe main text.
9o Pauli blocking where ξ ∼ q/µ F with q the the momentum transfer during the scatteringand µ F the neutron chemical potential.In the equilibrium epoch, t (cid:29) τ eq , the total annihilation rate Γ A = C c ∝ σ χn . When ˜ ε Z is at the resonant point, σ χn is enhanced drastically by the factor m Z / Γ Z so does the DMheating resulted from DM emissivity (cid:15) χ . This accounts for the dip at m V = m Z in eachfigure.On the other hand, DM heating for m χ in the sub-GeV region is much stronger. It canbe understood that, as m χ (cid:28) m n , q ∝ m χ while m V /m χ is held fixed, we have σ χn ∝ ˜ ε Z /m χ according to Eq. (10). Hence a smaller m χ leads to a larger σ χn as well as a more effectiveDM heating. However, the effect of DM heating will not grow indefinitely with ˜ ε Z as σ χn ≤ σ geom χn ≈ − cm . The maximum T s caused by DM heating saturates when σ χn = σ geom χn and is around 3100 K. This justifies our numerical results in Fig. 2 that T s does not increasefurther when ˜ ε Z is sufficiently large for a given m χ .To all plots in Fig. 2, DM heating becomes weaker instead of proportional to 1 /m χ for m χ (cid:46) O (170) MeV. Although DM is capable of producing e ± and µ ± in this mass range,the chemical potentials for both particles are µ eF ∼ O (170) MeV and µ µF ∼ O (70) MeV.All channels are Pauli blocked and only pions formed by q ¯ q are allowed until m χ < m π .Nonetheless, in the presence of V − Z mass mixing, neutrinos are also part of the annihilationproducts and take a significant branching ratio in the DM annihilation at such a mass region.But neutrino cannot contribute to the heating. This explains why T s is much colder when m χ (cid:46) O (170) MeV. The DM heating in this region is mainly due to kinetic heating. As σ χn = σ geom χn , the resulted T s is around 1750 K from pure kinetic heating.Various η values in Fig. 2 characterize the contributions from ε γ,Z to ˜ ε Z . Both η = 1and 0 are similar because even ε γ (cid:54) = 0, its effect to ˜ ε Z is suppressed by m V /m Z as seenfrom Eq. (A5). For η = 1, the kinetic mixing can contribute comparably to the V − Z massmixing unless m V > m Z . This can be clearly seen in Fig. 2 that the difference between η = 1 and 0 is apparent only in m V > m Z region, which is the region to the right of the dip. We found that the equilibrium condition holds in most of the parameter space in this work. However, inthe calculation we adopted Γ A = C a N χ with N χ given by Eq. (6), instead of simply assuming Γ A = C c . Since neutron is charge neutral, Q = 0, the effect of kinetic mixing Qε γ in Eq. (A4a) has zero contributionto σ χn . Hence σ χn ∝ ˜ ε Z . Nonetheless, if protons in NS are considered, then ε γ shall contribute to theDM-proton cross section σ χp as a consequence of non-vanishing Qε γ .
10o the left of the dip, the contribution from ε γ to ˜ ε Z for η = 1 is negligible.For η = ∞ , ε Z vanishes so that the only contribution to ˜ ε Z comes from ε γ . As discussedearlier, the effect of kinetic mixing term is suppressed by m V /m Z and thus σ χn ∝ ˜ ε Z ∝ ε γ m V /m Z . The associated DM heating is in general much weaker than the cases with η = 1and 0. However, the advantage of η = ∞ is that no neutrino can be produced in theDM annihilation due to the absence of V − Z mass mixing. The energy released from DMannihilation can be fully deposited into NS. This accounts for the higher T s than η = 1 and0 in terms of the same σ χn . But the difference is not apparent. Numerical calculation showsit is around tens to O (100) K. B. Case for m V < m χ For light mediator case, the channel χ ¯ χ → V dominates over χ ¯ χ → f ¯ f due to α χ (cid:29) ε γ,Z in general. As long as F ∼ V can fully decay into SM particles before it exits the NS.The resulting heating from χ ¯ χ → V with V → f ¯ f can be appreciable as shown in therightmost panel of Fig. 2. The heating region in the m V < m χ case is much more expandedthan the m V (cid:29) m χ case since lighter m V induces larger σ χn as shown in Eq. (10). Theresulting effects from different η ’s are similar to those in the previous subsection. WhenDMs mainly annihilate to 2 V , the thermal relic cross section is controlled by α χ and m χ while it is independent of ε γ,Z . Hence the thermal relic cross section only constrains m χ when α χ and m V are fixed. For α χ = 1 and 0 .
01, the m χ values rendering the thermal reliccross section are around 2 × MeV and 5 × MeV, respectively.
V. DETECTABILITY OF THE FUTURE TELESCOPE
Since DM annihilation could significantly affect the NS surface temperature T s , we discussthe detectability of T s in the JWST and similar telescopes in the future. The blackbodyspectral flux density with T s at a given frequency ν is given by [20] f ν ( ν, T s , d ) = 4 π ν e πν/k B T s − (cid:18) R γd (cid:19) (11)where d is the distance between the NS and the Earth. Taking T s = 2000 K and d = 10 pcas an example, we have f ν ≈ .
84 nJy at ν − = 2 µ m. The signal-to-noise ratio (SNR)11 �� �� ������������������������� � [ �� ] � � [ � ] ��� = � ��������� � �� �� � � �� FIG. 4: The exposure time t exp for SNR = 2 in JWST. The region enclosed by the red lineindicates t exp ≤ s.for JWST-like telescope is proportional to f ν √ t exp where t exp is the exposure time. FromRef. [58], JWST covers 0 . µ m to 2 . µ m imaging sensitivity in its Near-Infrared Imager(NIRI). A F200W filter centered at ν − = 2 µ m reaches SNR = 10 with f ν = 10 nJy and t exp = 10 s.In Fig. 4, we plot the t exp for obtaining SNR = 2 over d − T s plane. The region enclosedby the red line represents t exp < s. There are multiple filters available for NIRI with ν − centered at various different values [58]. We select the filter with ν − most suitablymatching the corresponding blackbody wavelength at T s . This explains the zigzag behaviorin Fig. 4. In principle, as σ χn ∼ σ geom χn , kinetic heating can maximally warm the NS up to1750 K without DM annihilation. For NS that is located within 10 pc, JWST can achieveSNR = 2 with t exp ≤ s for T s ≥ VI. SUMMARY AND OUTLOOK
In this work we have investigated the new dynamics arising from the kinetic mixingand V − Z mass mixing between the dark gauge boson V of the broken U (1) X symmetryand neutral gauge bosons in SM. In particular, V − Z mass mixing induces a resonanceat m V ≈ m Z , which can be seen from the pole of ˜ ε Z at m V = m Z . The axial-vector partof the coupling between V and SM fermions has been included in our calculations. As12 χ χnn FIG. 5: Feynman diagram for DM-neutron scattering. The blob is an effective vertex thatincludes both vector and axial-vector contributions from kinetic mixing and V − Z massmixing. χ ¯ χ → V dominates the annihilation channel for m V < m χ , V can decay into a pair ofSM fermions before it exits NS and induce NS heating in addition to χ ¯ χ → f ¯ f . Althoughthis contribution appears naturally in the dark boson model considered here, it is usuallynot included in the model-independent analysis, such as the one performed in Ref. [22]. Wealso demonstrated numerically that NS can provide constraints on sub-GeV DM with feeblecoupling to SM particles in complementary to the current direct search. The detectabilitywith reasonable t exp in JWST telescopes is discussed. Similar conclusion can be drawn forthe future JWST-like telescopes.We note that this work only considers χn scattering in the capture rate. This explainswhy NS is not sensitive to the dark sector when ε Z = 0 ( η = ∞ ). Neutron interacts with DMonly through NC interaction governed by ˜ ε Z . Once ε Z = 0, NC interaction becomes muchsuppressed since ε γ in ˜ ε Z is oppressed by m V /m Z . However, NS also consists of protonsalthough the fraction of them is rather small. When protons are included, CC interactionwill be involved for the capture of DM and NS remains sensitive to the dark sector even for ε Z = 0. In general, NS sensitivity will be improved by including proton contributions. Weleave this for future studies. Appendix A: DM-neutron interaction
When DMs fall into NS, they could scatter with neutrons via exchanging the dark boson V as shown in Fig. 5. The kinetic mixing and V − Z mass mixing generate vector and axial-vector interactions between V and SM fermions. The usual derivation of these interactions13roceeds through the diagonalization of both L gauge and L mass in Eqs. (2) and (3), whichgives rise to relations between fields in the gauge basis and those in mass eigenstate basis.However, since we are only interested in interactions up to O ( ε γ ) or O ( ε Z ), we do not needto perform the diagonalization but rather treating the mixing terms ε γ B µν V µν / (2 cos θ W )and ε Z m Z Z µ V µ as perturbations. These two mixing terms generate the following two-pointfunctions at the tree level i Π µνV γ = iε γ k g µν ,i Π µνV Z = − i ( ε γ tan θ W k + ε Z m Z ) g µν , (A1)where k is the four-momentum of V entering into kinetic mixing or V − Z mixing vertex.Hence the EM coupling of V to SM fermions results from multiplying the two-point function i Π µνV γ , the photon propagator iD γαµ ( k ), and the electromagnetic coupling ieA α J α EM , as shownin Fig. 6. This multiplication leads to ieJ α EM − ig αµ k iε γ k g µν V ν = ieε γ J ν EM V ν . (A2)Similarly, NC coupling of V to SM fermions is given by multiplying the two-point function i Π µνV Z , the Z boson propagator iD Zαµ ( k ), and the NC coupling igZ α J α NC / cos θ W . This givesrise to ig cos θ W J α NC − ik − m Z + im Z Γ Z (cid:18) g αµ − k α k µ m Z (cid:19) ( − i )( ε γ tan θ W k + ε Z m Z ) g µν V ν = − ig cos θ W J ν NC V ν ( ε γ tan θ W m V + ε Z m Z )( m V − m Z + im Z Γ Z ) . (A3)Here we have used the physical conditions k = m V and k µ (cid:15) µV = 0. We have also chosenunitary gauge for the Z boson propagator. Therefore, the interaction vertex between darkboson and neutron in Fig. 5 has the following Lorentz structure ie ¯ ψ n γ µ ( a f + b f γ ) ψ n with a f = Qε γ + 1sin 2 θ W ( I − Q sin θ W )˜ ε Z , (A4a) b f = − I sin 2 θ W ˜ ε Z , (A4b)where ˜ ε Z = ε Z + ε γ tan θ W ( m V /m Z )(1 − m V /m Z ) + Γ Z /m Z (cid:18) − m V m Z − i Γ Z m Z (cid:19) (A5)and Γ Z is the Z boson decay width, Q and I are the electric charge and the weak isospinrespectively. In Tab. I, we list Q and I for various particles. The values for neutron can be14 ¯ ff = V γ, Z f ¯ f FIG. 6: Feynman diagrams contributing to the coupling of dark boson V to SM fermions. u d c s t b (cid:96) νQ −
13 23 −
13 23 − − I −
12 12 −
12 12 − −
12 12
TABLE I: Values of Q and I for quarks, leptons and neutrinos.obtained by summing the corresponding quantum numbers of three quarks udd in the lowenergy limit.Mixing parameters ε γ and ˜ ε Z are responsible for EM and NC interactions, respectively.EM interaction does not contribute to σ χn since Q = 0 for neutron. On the other hand ˜ ε Z has a feeble dependence on ε γ with a suppression factor m V /m Z when m V (cid:28) m Z . Thisexplains why σ χn is still nonzero when ε Z = 0 ( η = ∞ ).The spin-averaged χn scattering amplitude is given by |M χn | = 8 πα χ ( t − m V ) {− m n [( b f − a f ) m χ + a f u + b f ( s + u )] + 2( a f + 3 b f ) m χ − a f um n + a f ( t + 2 tu + 2 u ) + 2( a f − b f ) m n + b f ( s + u ) } , (A6)where s , t and u are the Mandelstam variables. DM scatters with neutrons with its velocityboosted to 0 . c − . c by the NS gravity. It must be treated relativistically. However,neutrons can be treated as at rest since its chemical potential is O (200) MeV in the star.Therefore, from the method in Ref. [72], we are able to write down the DM-neutronscattering cross section as σ χn = 116 πλ / ( s, m , m ) λ / ( s, m , m ) (cid:90) t + t − |M χn | dt (A7)where λ ( x, y, z ) = x + y + z − xy − yz + 2 xz (A8)15s the K¨all´en function, t ± = 12 (cid:88) i =1 m i − s − s ( m − m )( m − m ) ± λ / ( s, m , m ) λ / ( s, m , m )2 s , (A9)and s = m + m + 2 E m , (A10)where m = m = m χ and m = m = m n for the χn scattering. In Eq. (A10), the energy E = γm is the total energy carried by particle 1, which is DM.
1. Pauli blocking in the χn scattering Note that if the momentum transfer √− t in Eq. (A7) is smaller than the Fermi mo-mentum, the suppression by Pauli blocking takes effect. We include this in the numericalcalculation by incorporating the method in Ref. [60]. Our result agrees with Ref. [60] in thethree benchmark scenarios that |M χn | are constant, t -dependent and t -dependent.
2. Axial-vector contribution in the NR limit If χ can be treated non-relativistically as well, we have s = m χ + m n + 2 m χ m n , u = m χ + m n − m χ m n and t = 0. Therefore the amplitude and the cross section become, |M NR χn | = 64 πa f α χ m χ m n m V (A11)and σ NR χn = 4 a f α χ m V m χ m n ( m χ + m n ) (A12)which are independent of b f where it determines the strength of axial-vector coupling. Appendix B: DM annihilation
We can divide the DM annihilation into two categories, which are m V ≥ m χ and m V
Dark bosons can be produced from DM annihilation once m V < m χ . This channel isthought to have feeble effect on the heating since V interacts with the NS medium weaklyand escapes without any trace. However, we found that, depending on the strength of ε γ,Z , V can decay into SM particles before it reaches the surface of the star. In the case thatthe decay length (cid:96) dec is much smaller than the star’s radius, the total energy released fromthe annihilation can be fully deposited to the star. See Fig. 1a. We also examine the casethat V is produced in the DM rich region in the star’s center. V could undergo multiplescattering with the surrounding DMs and self-trapped until it decays. See Fig. 1b. This isanother way to extract energy from V . We discuss both effects in the following.18 � � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � η = � �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε γ η = ∞ ���������������� � FIG. 8: Fraction F of dark boson decay into SM particles that contributes to the heatingeffect.
1. Decay length
The dark boson decay length with time dilation effect is given by (cid:96) dec = vγτ dec , (C1)where v = (cid:113) − m V /m χ is the V velocity and τ dec = Γ − V the V lifetime at rest. Let usassume that V is produced in the center of the star and its propagation distance is R .Fig. 8 presents F defined in Eq. (9), i.e., the fraction of V converting into SM particles aftertraveling a distance r = R , as functions of ε Z,γ and m χ for m V = 0 . m χ . We have subtractedneutrino contributions from F since they cannot generate heat. Since the branching ratio of V decays to neutrinos is nonzero in the case of V − Z mass mixing, F is generally smallerthan 1 for η = 1. For η = ∞ , no neutrinos can be produced, thus F can reach unity.In these figures, the chemical potential for electron µ eF is about O (170) MeV. For a darkboson at rest with m V ≤ µ eF , V → e + e − can be Pauli blocked even for m V ≥ m e . On theother hand, if V is highly boosted as a result of heavy DM annihilation, V → e + e − is notPauli blocked as long as m χ ≥ µ eF . Therefore, to enable V → f ¯ f decays, two conditions arerequired. The first is m χ ≥ µ fF and the second is m V ≥ m f .19 V χV V χVχ
FIG. 9: χV scattering via s and t channels.
2. Dark boson-DM interaction length
Feynman diagrams contributing to χV scattering are shown in Fig. 9 and the amplitudeis given by |M χV | = 64 π α χ ( s − m χ ) ( t − m χ ) { m V [6 m χ ( s + t ) − m χ + s − st + t ] − m χ (3 s + 14 st + 3 t ) + m χ ( s + 7 s t + 7 st + t )+ 4 m V [ m χ ( s + t ) − stm χ + st ( s + t )] + 6 m χ − st ( s + t ) } . (C2)To compute the scattering cross section σ χV , it is fair to assume DM at rest. However, V is produced with relativistic velocity since m χ > m V . We follow the procedure given inEqs. (A7)-(A10) and set m = m = m V and m = m = m χ . Thus, σ χV = 116 πλ ( s, m V , m χ ) (cid:90) t + t − |M χV | dt. (C3)Note that χV scattering is not subject to Pauli blocking since DMs do not become degeneratein the presence of annihilation.The χV scattering length (cid:96) χV is given by (cid:96) χV = ( n χ σ χV ) − , (C4)with n χ ≡ N χ /V χ the average DM number density. The volume characterizing DMs in NSis V χ = 4 πr / r th ≈ . × cm (cid:18) T χ K 10 MeV m χ (cid:19) / (C5)is the thermal radius. If (cid:96) χV (cid:28) r th , V can scatter with surrounding DMs multiple timesand gradually lose its kinetic energy. However, our numerical result shows that (cid:96) χV (cid:29) r th � � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε � η = � �� � �� � �� � �� � �� � �� - �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � � χ [ ��� ] ε γ η = ∞ ����� � �� �� ℓ χ � / � � � FIG. 10: The ratio (cid:96) χV /r th for η = 1 and ∞ . We take α χ = 1 and T χ = 1000 K in thecalculation.in all of our interested parameter space. In Fig. 10, we take α χ = 1 and T χ = 1000 K. Thechoice α χ < (cid:96) χV even longer due to a weaker χV interaction. For η = 1, the regionfor (cid:96) χV /r th < m χ (cid:46)
300 MeV . However, even V can be self-trapped, ithardly decays into particles other than neutrinos because the allowed channels, eg. e ± and µ ± , are Pauli blocked. For η = ∞ , only a very small parameter space leads to (cid:96) χV /r th < V is insignificant, hence only V decayscontribute to the energy injection. [1] G. Aad et al. [ATLAS Collaboration], Eur. Phys. J. C , 299 (2015) [Erratum ibid , 408(2015)] [arXiv:1502.01518 [hep-ex]].[2] J. Abdallah et al., Phys. Dark Univ. , 8 (2015) [arXiv:1506.03116 [hep-ph]].[3] J. Aalbers et al. [DARWIN Collaboration], JCAP , 017 (2016) [arXiv:1606.07001 [astro-ph.IM]].[4] D. S. Akerib et al. [LUX Collaboration], Phys. Rev. Lett. , 021303 (2017) [arXiv:1608.07648[astro-ph.CO]].[5] C. Amole et al. [PICO Collaboration], Phys. Rev. Lett. , 251301 (2017) [arXiv:1702.07666[astro-ph.CO]].
6] D. S. Akerib et al. [LUX Collaboration], Phys. Rev. Lett. , 251302 (2017) [arXiv:1705.03380[astro-ph.CO]].[7] E. Aprile et al. [XENON Collaboration], Phys. Rev. Lett. , 181301 (2017)[arXiv:1705.06655 [astro-ph.CO]].[8] E. Aprile et al. [XENON Collaboration], Phys. Rev. Lett. , 111302 (2018)[arXiv:1805.12562 [astro-ph.CO]].[9] E. Aprile et al. [XENON], Phys. Rev. Lett. , 241803 (2019) [arXiv:1907.12771 [hep-ex]].[10] E. Aprile et al. [XENON], Phys. Rev. Lett. , 251801 (2019) [arXiv:1907.11485 [hep-ex]].[11] M. G. Aartsen et al. [IceCube PINGU Collaboration], arXiv:1401.2046 [physics.ins-det].[12] K. Choi et al. [Super-Kamiokande Collaboration], Phys. Rev. Lett. , 141301 (2015)[arXiv:1503.04858 [hep-ex]].[13] M. G. Aartsen et al. [IceCube Collaboration], Eur. Phys. J. C , 146 (2017) [arXiv:1612.05949[astro-ph.HE]].[14] M. Aguilar et al. [AMS Collaboration], 211101 (2015).[15] M. Ackermann et al. [Fermi-LAT Collaboration], Astrophys. J. , 43 (2017)[arXiv:1704.03910 [astro-ph.HE]].[16] G. Ambrosi et al. [DAMPE Collaboration], Nature , 63 (2017) [arXiv:1711.10981 [astro-ph.HE]].[17] C. Kouvaris, Phys. Rev. D , 023006 (2008) [arXiv:0708.2362 [astro-ph]].[18] A. de Lavallaz and M. Fairbairn, Phys. Rev. D , 123521 (2010) [arXiv:1004.0629 [astro-ph.GA]].[19] C. Kouvaris and P. Tinyakov, Phys. Rev. D , 063531 (2010) [arXiv:1004.0586 [astro-ph.GA]].[20] M. Baryakhtar, J. Bramante, S. W. Li, T. Linden and N. Raj, Phys. Rev. Lett. , 131801(2017) [arXiv:1704.01577 [hep-ph]].[21] N. Raj, P. Tanedo and H. B. Yu, Phys. Rev. D , 043006 (2018) [arXiv:1707.09442 [hep-ph]].[22] C. S. Chen and Y. H. Lin, JHEP , 069 (2018) [arXiv:1804.03409 [hep-ph]].[23] N. F. Bell, G. Busoni and S. Robles, JCAP , 018 (2018) [arXiv:1807.02840 [hep-ph]].[24] J. F. Acevedo, J. Bramante, R. K. Leane and N. Raj, arXiv:1911.06334 [hep-ph].[25] A. Joglekar, N. Raj, P. Tanedo and H. B. Yu, arXiv:1911.13293 [hep-ph].[26] W. Y. Keung, D. Marfatia and P. Y. Tseng, JHEP , 181 (2020) [arXiv:2001.09140 [hep-ph]].[27] C. Kouvaris and P. Tinyakov, Phys. Rev. D , 083512 (2011) [arXiv:1012.2039 [astro-ph.HE]].
28] S. C. Leung, M. C. Chu and L. M. Lin, Phys. Rev. D , 107301 (2011) [arXiv:1111.1787[astro-ph.CO]].[29] C. Kouvaris, Phys. Rev. Lett. , 191301 (2012) [arXiv:1111.4364 [astro-ph.CO]].[30] S. D. McDermott, H. B. Yu and K. M. Zurek, Phys. Rev. D , 023519 (2012) [arXiv:1103.5472[hep-ph]].[31] T. G¨uver, A. E. Erkoca, M. Hall Reno and I. Sarcevic, JCAP , 013 (2014)[arXiv:1201.2400 [hep-ph]].[32] J. Bramante, K. Fukushima and J. Kumar, Phys. Rev. D , 055012 (2013) [arXiv:1301.0036[hep-ph]].[33] J. Bramante, K. Fukushima, J. Kumar and E. Stopnitzky, Phys. Rev. D , 015010 (2014)[arXiv:1310.3509 [hep-ph]].[34] C. Kouvaris and P. Tinyakov, Phys. Rev. D , 043512 (2014) [arXiv:1312.3764 [astro-ph.SR]].[35] M. I. Gresham and K. M. Zurek, Phys. Rev. D , 083008 (2019) [arXiv:1809.08254 [astro-ph.CO]].[36] B. Grinstein, C. Kouvaris and N. G. Nielsen, Phys. Rev. Lett. , 091601 (2019)[arXiv:1811.06546 [hep-ph]].[37] R. Garani, Y. Genolini and T. Hambye, JCAP , 035 (2019) [arXiv:1812.08773 [hep-ph]].[38] G. L. Lin and Y. H. Lin, JCAP , 022 (2020) [arXiv:2004.05312 [hep-ph]].[39] A. Nelson, S. Reddy and D. Zhou, JCAP (2019), 012 [arXiv:1803.03266 [hep-ph]].[40] J. Ellis, G. H¨utsi, K. Kannike, L. Marzola, M. Raidal and V. Vaskonen, Phys. Rev. D (2018), 123007 [arXiv:1804.01418 [astro-ph.CO]].[41] A. Bauswein, G. Guo, J. H. Lien, Y. H. Lin and M. R. Wu, [arXiv:2012.11908 [astro-ph.HE]].[42] R. K. Leane, T. Linden, P. Mukhopadhyay and N. Toro, [arXiv:2101.12213 [astro-ph.HE]].[43] S. Tulin and H. B. Yu, Phys. Rept. , 1 (2018) [arXiv:1705.02358 [hep-ph]].[44] S. W. Randall, M. Markevitch, D. Clowe, A. H. Gonzalez and M. Bradac, Astrophys. J. ,1173 (2008) [arXiv:0704.0261 [astro-ph]].[45] M. G. Walker and J. Penarrubia, Astrophys. J. , 20 (2011) [arXiv:1108.2404 [astro-ph.CO]].[46] M. Boylan-Kolchin, J. S. Bullock and M. Kaplinghat, Mon. Not. Roy. Astron. Soc. , L40(2011) [arXiv:1103.0007 [astro-ph.CO]].[47] M. Boylan-Kolchin, J. S. Bullock and M. Kaplinghat, Mon. Not. Roy. Astron. Soc. , 1203(2012) [arXiv:1111.2048 [astro-ph.CO]].
48] O. D. Elbert, J. S. Bullock, S. Garrison-Kimmel, M. Rocha, J. O˜norbe and A. H. Peter, Mon.Not. Roy. Astron. Soc. , 29 (2015) [arXiv:1412.1477 [astro-ph.GA]].[49] B. Holdom, Phys. Lett. , 196 (1986)[50] P. Galison and A. Manohar, Phys. Lett. , 279 (1984)[51] R. Foot, Int. J. Mod. Phys. D , 2161 (2004) [astro-ph/0407623].[52] D. Feldman, B. Kors and P. Nath, Phys. Rev. D , 023503 (2007) [hep-ph/0610133].[53] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer and N. Weiner, Phys. Rev. D , 015014(2009) [arXiv:0810.0713 [hep-ph]].[54] M. Pospelov and A. Ritz, Phys. Lett. B , 391 (2009) [arXiv:0810.1502 [hep-ph]].[55] K. S. Babu, C. F. Kolda and J. March-Russell, Phys. Rev. D , 6788 (1998) [hep-ph/9710441].[56] H. Davoudiasl, H. S. Lee and W. J. Marciano, Phys. Rev. D , 115019 (2012) [arXiv:1203.2947[hep-ph]].[57] H. Davoudiasl, H. S. Lee, I. Lewis and W. J. Marciano, Phys. Rev. D , no. 1, 015022 (2013)[arXiv:1304.4935 [hep-ph]].[58] J. P. Gardner, J. C. Mather, M. Clampin, R. Doyon, M. A. Greenhouse, H. B. Hammel,J. B. Hutchings, P. Jakobsen, S. J. Lilly and K. S. Long, et al. Space Sci. Rev. , 485(2006) [arXiv:astro-ph/0606175 [astro-ph]] and JWST pocket guide[59] W. Skidmore et al. [TMT International Science Development Teams & TMT Science AdvisoryCommittee], Res. Astron. Astrophys. , 1945-2140 (2015) [arXiv:1505.01195 [astro-ph.IM]].[60] N. F. Bell, G. Busoni, S. Robles and M. Virgato, JCAP , 028 (2020) [arXiv:2004.14888[hep-ph]].[61] N. F. Bell, G. Busoni, S. Robles and M. Virgato, [arXiv:2010.13257 [hep-ph]].[62] J. Bramante, A. Delgado and A. Martin, Phys. Rev. D (2017), 063002 [arXiv:1703.04043[hep-ph]].[63] B. Dasgupta, A. Gupta and A. Ray, JCAP (2019), 018 [arXiv:1906.04204 [hep-ph]].[64] E. M. Riordan, M. W. Krasny, K. Lang, P. De Barbaro, A. Bodek, S. Dasu, N. Varelas,X. Wang, R. G. Arnold and D. Benton, et al. Phys. Rev. Lett. , 755 (1987)[65] A. Bross, M. Crisler, S. H. Pordes, J. Volk, S. Errede and J. Wrbanek, Phys. Rev. Lett. ,2942 (1991)[66] M. Abdullah, J. B. Dent, B. Dutta, G. L. Kane, S. Liao and L. E. Strigari, Phys. Rev. D ,015005 (2018) [arXiv:1803.01224 [hep-ph]].